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"Without the AP, the Everything is indeed devoid of any
structure whatsoever."
I believe it is incorrect to attribute any fixed condition to the
Everything. This includes its structure. The Everything to be
informationless - that is absent any selection - would need to have an ever
changing random structure. This random shifting produces part of the
dynamic that drives the evolution of some types of universes [those
allowing true noise] from state to state and drives universes from type to
type.
Hal